Holy Family Catholic High School. By Mr. Gary Kannel, Mathematics Teacher. Version 0.3 Last Modified 4/04/2008

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1 Holy Family Catholic High School Algebra Review By Mr. Gary Kannel, Mathematics Teacher Version 0. Last Modified /0/008 Restrictions: Although you may print a copy for your personal use in reviewing for the placement test, this review is not to be distributed in any form. This is intended as a supplement to your regular schoolwork and may not be used by a teacher in a classroom environment.

2 This packet will review some of the topics needed on the Holy Family Catholic High School mathematics placement tests. It is by no means complete nor should it be considered a substitute for taking an actual high school level Algebra I course. It is intended only for those that completed or are at least close to completing an Algebra I course. At times it may seem like it is insultingly easy but stick with it and I believe you will learn something. One thing I also want to be clear about is that solving a problem is not the same as just getting the correct answer. That is like saying the only important thing about a roller coaster ride is where it ends. It really is how you get there that matters most. Okay I won t pretend that anyone reading this will think that solving a math problem is as much fun as a roller coaster though, perhaps, to some of you it is just as terrifying in that case, perhaps I can help. Anyway, please pay attention to how the problem is solved and how the work is shown. When it finally does come to the test, you will be graded not just on the correct results but also how you solved your problems. That does not mean that it is do it my way or you automatically lose points. There are frequently multiple methods of solving a problem just make sure yours is a valid algebraic method and not just something else like guess-n-check.

3 Unit : Conversions and Things Time to get started but before we get to real algebra we will start with some basics which, by the way, you need to be able to do without a calculator. Rounding Frequently (especially in the real world aka not school) answers to problems do not work out evenly. A lot of the time we really need the eact answer so answers are stuck being messed up disasters like. Have you ever tried cutting a piece of paper to be inches long? Come on - Good luck! When the problem and situation calls for it use rounding. Of course you are familiar with the concept but some people still make mistakes with it. For eample, you could probably handle rounding to say decimal places as in the following You know the basic rules. For eample, if the digit after what you want is a or higher then round up, otherwise round down. The question now is, were you aware that.0 rounded to decimal places is not.. It is actually.0. I know you are probably thinking, what is the difference? If they were eact numbers than there wouldn t be any but as rounded off answers there is one.. would be from rounding off to two decimal places not three in other words less accurate. Basically if the problem asks for a certain number of decimal places (unless it works out eactly to fewer digits) keep the zeroes there. Now some problems for you to do (don t blame me I made this to be an optional packet) Round each of the following to the given number of decimal places P to decimal place P to decimal places (careful watch those zeroes) P -. to 0 decimal places (or nearest integer same thing) P -.06 to decimal places. Now it is time for number conversions. We are frequently given numbers in an inconvenient form, or the natural process of solving problems generates less-than-desirable (even ugly) results. These net sections are about being able to convert from one form of a number to another. Fractions Decimals This is probably the easiest conversion. A fraction is simply another way of stating a 6 division. is the same as saying 6. Both forms are equal to. So in order to convert a fraction, all you have to do is divide the top number by the bottom number. The thing to be aware of is that the conversion may not be always work out evenly. Sometimes when you try, it just won t end so you need to round. Look to see if the problem specified a certain number

4 places. For eample trying 7 on a calculator will yield. the eact number of threes varies by calculator since in reality is goes on forever though there is an obvious pattern. You will later learn that any fraction can either be written as a terminating decimal (written eactly with set number of decimal places) or repeating (going on forever but it is just repeating the same sequence of numbers over and over again) Note that you may have heard of numbers like which go on forever but do not ever repeat. and numbers like it cannot be written as a fraction; hence, they are referred to as irrational numbers (cannot be written as a ratio of other numbers). Eample a Convert to a decimal rounded to decimal places Solution:.6 Convert the fraction to decimal (round to decimal places) P - P -6 / P -7 P -8) 9 Decimals Fraction We will focus (for now) only on the terminating decimals such as.. These are relatively straightforward. First, figure out what place the last non-zero digit is in. In the case of. it is the tenths place. Then, place the shown number over that number so in this case although with that done you do need to try to reduce the fraction. Note that since evenly divides both and 0 we can reduce as follows 0 0 Here is another eample..76 has the two in the thousands place so Convert the decimal to a fraction (remember that you always reduce fractional answers) P -9. P -0. P P -.09 Improper Fraction Mied Number Improper fractions are those that have a larger number on top (the numerator) than on the bottom (the denominator). To convert a number like 7 you follow this process. First, divide

5 the 7 by. Second, write that integer part (or whole part) in front of the fraction. The remainder of that division (the leftover part - in this case, a since only equals 6 so 7 has an etra ) goes in the new numerator. The denominator stays the same. So, the result in this case is Here is another eample: First note that when dividing 0 by we get an 8 but since 8 is only 0 there are still left over which means: 8 Convert the improper fraction to a mied number 9 P - P - P - 7 P -6 0 Mied Numbers Improper Fraction Mied numbers are generally only useful as final answers. While solving, it is often easier to deal with a number when it is an improper fraction. For eample, it is impossible to deal with the reciprocal of a number (useful when dividing) without first thinking of it as fraction (improper or otherwise). Say that you have and want the improper fraction version. Multiply the integer part (the ) by the denominator () and then add the numerator (). Place the result over the denominator. In other words 9 Another eample would be 7 convert the mied number to an improper fraction P -7 P -8 P P

6 Decimal Percent This conversion merely requires moving the decimal point places to the right. An eample would be.67 = 6.7% The mistake most commonly made is to say that. = % when in fact. = 0% Convert the decimal to percent P -.6 P -.08 P -. P Percent Decimal This is the opposite of the last topic. For eample, changing the percentage 7.89% into decimal.789. The key here is to still make sure that you moved the decimal point two places but now move it to the left. The only difference is you need to move it left. You might, for eample, say that.% =. when it should be.% =.0%. Another version of this mistake is to say that % =. when in fact % =. Convert the percent to decimal P -.% P % P -7 % P -8.% Eponents All eponents are is a shortcut for writing out what could be long strings of multiplications. For eample could have just been written... The first format is just a lot easier to write (especially if I said 00 which would otherwise require writing out a hundred s and quite frankly I don t have that kind of time). Before I start mentioning some of the quirks and basic facts, I should start with some terminology. Let s use as a reference. In the epression, the is known as the base. It is that thing which has the eponent or that number which is being multiplied repeatedly. Whatever you want to think of it as, just know that the big number is called the base. The other one which is written as a superscript (a fancy word but which breaks down easily. super = above and script = written so superscript just means written above) is just called the eponent (like that wasn t obvious) The eponent, in this case, reflects the number of times the number of times the number will be multiplied. It might be useful to review what the true definition of an eponent is. y means to multiply by, y times. In reality then,. The difference seems minor since

7 putting the one in front of the s makes no difference in the value, but suppose we were to try 0. Saying this is just no s makes very little sense because what number is it then. Now we can think of this as 0. It is a multiplied by zero times. By putting the one in front of every eponent epansion we do not need to consider the 0 eponent as a different rule. Confused yet? You probably are but bear with me for a little while longer. Look at the following table Each time you increase the eponent by one you are simply multiplying the previous result by the base. Since it is only logical that = the question becomes: what times equals the desired number. Well is the only valid answer. If we tried saying that 0 = 0 we would have to conclude that = 0. = which is really messed up. Now that we have established the rule that each time the eponent is increased by the previous answer is multiplied by the base we can consider the other direction. Our pattern (notice I said Our pattern I think it is supposed to make you feel like we are working together. Doesn t work does it?) Anyway, our pattern (the one where =. ) would suggest that 0 = -.. Since the answer is supposed to be we are forced to ask what times equals. (. =) the only solution is that - = ½. This process yields the following table - ¼ / 8 - ½ / ¼ - ½ ½ You will hopefully have noticed the following 8 It seems that negative eponents evaluate just like their positive counterparts, ecept the results are now placed in the denominator under a numerator of. By the way, realize that it usually does not matter what the base is when the eponent is 0. The result is nearly always. The only eception is 0 0. Logically, 0 to any power should be 0 but by our definition it is also true that anything to the 0 should be. 0 0 can not equal two things at once so we are forced to conclude that 0 0 is undefined. (That is the mathematician s way of saying I can t make any sense of this either so I am giving up.) This last section included some technical discussion and was probably confusing so I suggest reading it a few more times. Perhaps read it twice before trying the homework and once more after or, perhaps, in about a week. Note that throughout this review I may give directions stating to use a certain number of nonzero decimal places. If I merely said decimal places I would find students rounding (correctly

8 I might add) answers like to just.000. While correct, this is not what I want. By saying non-zero decimal places my intention is to have the first three zeroes ignored and the counting to begin with the. The answer is therefore.000. Note that if a zero appears between numbers it should be counted. Only ones that do not follow a different number are skipped. So here are some answers given the way we want Please get in the habit of using the approimately symbol whenever you are rounding off an answer. Also did you notice that I included the trailing zero in the last problem but did not add any in the one before it. Think back to the bit on rounding. It is necessary to have it there. Writing the answer as just. in a section requiring places would have indicated that the answer was in fact eactly. when the answer in truth had to be rounded off. Evaluate the given eponents do this without a calculator since you may be required to do so on the test and or quiz P -9 P -0 6 P - - P - 8 P - - P - 7 P - 9 P

9 Unit : Absolutely Crazy I am not going to spend much time on absolute value. Absolute value is just the function that turns whatever is inside into a non-negative number. To zero and positive numbers, the absolute value bars act just like parenthesis. Eample: = ()= and 0 = (0) = 0. Just remember to not apply the absolute value function to components inside the absolute value. It is only the result of everything inside that is made non-negative. To show you what I mean - here is a common error: + - = + = 7. This is not repeat - not correct. The correct simplification is + - = - =. Another error is to make the final result of the entire problem positive. For eample: - - = - - = -. = - = which is again WRONG. The correct process yields = - - = -. = -. The final answer can be negative. It is only the inside of the absolute value that is forced to be a positive result (or zero let s not forget zero). Addition and subtraction The first part of this review is about addition and subtraction. Those ideas are so basic that you may have forgotten how you learned them. Think back to first and second grade. You probably had a number line posted on the wall. That was how you learned addition and subtractions so we might as well go back to that. Let s use the following. Pretend that each number is equally spaced out on the line (its pretty close anyway). Say that you were given the problem of + That means that you start out standing at and then take three steps to the right. You would find yourself at 7 so + = 7. I know this is simple but just stay with me -- unless you have a headache in which case you should probably skip the net paragraphs. In general, we always think of the addition of two numbers (a + b) as starting out at the first number (a) and then taking b steps to the right. This works even if we throw in problems such as + 8. We start at and take 8 steps to the right to end up at. (in other words + 8 = ) Now we need to deal with things like It is possible to always think of a negative as meaning the opposite direction. We are either starting in the opposite direction from 0 or we are moving in the opposite direction from normal addition which is to the right. This means that means to start at 8 and then instead of moving to the right we move to the left. When you move to the left you end up at so =. Of course, you may have noticed that I have only talked about addition. That is because, in one way of looking at things, subtraction is not a different operation. Technically subtraction is defined as a b = a + -b so subtraction is just a way to write an addition problem.

10 IMPORTANT: Note that you should also learn not to think of a - as a negative sign. It is either a subtraction symbol, when placed between two symbols or an opposite sign when placed in front of a number or variable. While this appears to be a subtle and perhaps insignificant difference, it is important. People frequently make a mistake because they think of -b as negative b when it really means the opposite of b. They assume that -b is a negative number when that is only true when b is positive to start with. If b is a negative number than - b is the opposite- a positive number. Back to subtraction, 9 means Simple enough but you could have handled that with your old way of thinking about subtraction. Look at a problem like You probably learned an arbitrary rule to handle this but since we now see subtraction as adding the opposite we just need to know that the opposite of is a positive to state the following: = 9 + (the opposite of ) = 9 + =. REALLY REALLY REALLY IMPORTANT: Yes, it is true that and there are times when the + - concept is very helpful while solving a problem; still, it is never acceptable as part of your final answer. If you are getting + - than your answer must be. Terminology sidebar: I referred to the opposite of a number earlier. This is somewhat sloppy terminology. It worked since I was dealing with the contet of addition. When you want opposite to mean things like and then the correct term is additive inverse. Think for a moment about what a number and its additive inverse add up to. An identity is a number, which, for that operation, returns the starting number. Using the a + b notation, what would b have to be in order for the sum to always be a. For eample, 7 + [] = 7. What needs to go in place of [] to make it work out? Hopefully, you know it is a 0. In fact a + 0 = a no matter what a is. This means that 0 is the additive identity. As it turns out, since 0 is its own additive inverse, it is also the subtraction identity, but that term is almost never used. When, in the last paragraph, I asked you to think about the addition result of a number and its additive inverse you, hopefully, realized that the result is always the additive identity, zero. Coincidence? Not really. P - What is the additive identity P - What is the additive inverse of P - What is the additive inverse of ½ P - How is subtraction defined? P - 0 P P -7-0 P -8 + P P -0 6

11 Multiplication I probably do not have to eplain what multiplication is but as long as I am eplaining everything else I might as well. In its most basic form multiplication means add the first number as many times as the second number. Eample 6 added si times 0 Of course this is only the base definition since it does a poor job eplaining how to handle 7 problems like Remembering that 7/ = /, It still makes very little sense to say this means but we will deal with fractions later. I just want to remind you of a few basic facts ) Communitive property a. b = b. a. =. ) Associative property (a. b). c = a. (b. c) (. ). =. (. ) ) Multiplication by zero a. 0 = 0 for any value of a. 0 = 0 ) Identity of multiplication = a. = a for any value of a. = You, hopefully, remember that we talked about identities earlier during addition. It was whatever, for the given operation, yields the original number. We also noted that adding the additive inverse yields the additive identity. That suggests that the multiplicative inverses should multiply to equal the multiplicative inverse,. What, then, is the multiplicative inverse of numbers like? What would need to replace [] in the equation. [] = to make it a valid equation? The only solution is what we call the reciprocal,. For any number, the multiplicative inverse or reciprocal is the fraction written with the numerator and denominator in the opposite positions. Numbers not written as fractions should just be thought of as over. So since since 7 7 Also remember the following general rules positive. positive = positive positive. negative = negative negative. positive = negative negative. negative = positive - - +

12 P - What is the multiplicative inverse of P - What is the multiplicative inverse of ½ Simplify P P P -. P P Matrices Matrices are a structured way to epress a collection of data (or equations as we will eventually learn in some other math class) in a simple bo form. For eample, I could list a price table as small t-shirt are $. and medium t-shirts are $.00 and large t-shirts are $.6 and small polo shirts are $6. and medium polo shirts are $6.7 and large polo shirts are $6.99. This same data could be written as small medium l arg e T shirt $.0 $.00 $.6 polo $6. $6.7 $6.99 You would probably agree that the matri form is a lot easier to deal with than the long list of prices. Imagine adding more categories of items and more sizes and the difference becomes even more obvious. There are times when matrices are simply data storage as in the previous eample but, depending on what information is stored there, it sometimes make sense to perform operations with them. For eample 9 0 The way to handle both addition and subtraction is to perform the operation on numbers that are in the same position first row first column to first row first column. So the previous is

13 Similarly Or However, what if I wanted you to do the following Those two matrices don t match which actually makes this a really easy problem all you do is say not possible. You can only add or subtract matrices when they are the same size. Simplify the following: P P P P Distributive property Originally, problems like (+) could not have anything done without knowing the actual value of. The distributive property lets us do something with it even though by itself we still cannot do much. The distributive property says that a(b + c) = ab + ac. Keeping in mind the definition of multiplication (ab = a +a + a.. + a b times or ab = b + b+b+...+b a times)

14 Or more concrete a( b c) ( b c) ( b c)..( b c) b c b c... b c b b.. b c c.. c ab ac ( ) a times each a times ( ) ( ) ( ) ( ) 0 This means that if there is a multiplication by a quantity inside parenthesis then it is possible just to multiply each part inside individually. By each part I mean each term separated by an addition or subtraction. It is not correct to say that (+)=.. +. =6+0 because the and are separated by multiplication. The correct simplification is (+)=8+0. Part of the power of the distributive property is actually in using it in the other direction. For eample since it is true to say that (+)=+6 we must also be able to say that +6=(+). So when faced with something like 0+ we can say that it is equal to 8(+) by using the distributive property although we often call this factoring. This property can often help us solve problems. It is not even limited simply to having numbers outside the parenthesis. It is also true that (+8) = = P P - 0 P - (+0) P - (-)+0 P -6 (6+) P ( ) P ( 0)

15 Unit : Fractured Fun Division Previously we dealt with addition and subtraction and finally multiplication. Division did not come up as a real issue (not counting that fraction conversion). In truth, the operation did come up in a different way. You may remember my saying that subtraction is merely addition (subtraction = addition of the additive inverse). In the same way, division may be defined as just a multiplication and, as before, by the inverse only this time it is the multiplicative inverse. The multiplicative inverse or reciprocal of a numbers, as it is usually called, is the number which when multiplied by the original number yields a product equal to one. For eample / is the reciprocal of since. / =. Of course as this eample shows, it is often easier to figure out the reciprocal than the technical definition might suggest. All you have to do is place it as the denominator of a fraction with as the numerator. Of course this would make no sense if the number started out as a fraction so here you just flip the number over numerator to denominator and vice-versa. Part of the importance of defining division this way (as opposed to a definition involving something like how many of b fit in a for a/b) is that now we can basically say that anything that works for multiplication also works for division (the primary eception being that 0 is a mess for division. It also is a great help when dividing by fractions. Consider the problem of. 7 Try figuring out how many five seventh s fit into three fourths and you will probably give yourself a headache. When division can be defined as multiplication by the reciprocal it is not that bad. You would solve it this way 7 the reciprocal of is so 7 A couple things to be wary of 7 7 Say that you are given a problem like 0 6, you may be tempted to say +. (hey 6 look s we have to be close to algebra) You hopefully are used to being able to reduce to but it does not work to do so in the original problem. Perhaps the best way to eplain why this does not work is to again use the definition of division. We will use the fact that the reciprocal of is /. 6 (6 ) (6 ) by definition of division 6 by the distributive property 6 = any number may be placed of

16 = multiplication of fraction The other thing is just how do deal with fractions so that is our net topic Fraction operations You have already been taught how to handle fractions, but this is only a review. If this quick lesson does not make sense than make sure you seek help soon. Learning to handle fractions can a big hurdle to get over but once you do, you will realize that they are not so bad Addition The key to fraction addition is common denominators. That is to say that the only way to add fractions together is if they have the same denominator. If you were putting together rolls of coins, you would want every coin in a roll to be the same type. If you try to put together a roll of quarters with dimes mied it, it would not work very well. It is about the same to try to do without a common denominator. Keep in mind that the denominator basically tells you what size pieces you have and the numerator is how many of those pieces you have. For eample, the can be thought of having taken a cake( or pizza if you prefer) which was divided into three parts of which you have. Since the other cake was divided into five pieces it is hard to say eactly how much you have since the pieces are different sizes. They do not fit together. But what if they were all one-sith of a cake. Imagine cutting each of the thirds in half. You would now have pieces all of which are the same size as if the cake had been cut into 6 pieces originally. Mathematically 6 The piece which was half a cake can be cut into three pieces which would yield pieces that are one-sith the size of the original cake. 6 So finally we can put is all together since all pieces are the same sizes Here is another eample without all the narration Hopefully you noticed a few things. First, that the basic process involved multiplying both the top and bottom by the same number. With top and bottom the same the fraction equals and as you already learned, multiplication by one does not change anything. Second, in the final addition step, I only added the numerators and left the denominator alone. A common mistake people make is to write 7/0 but you do not add denominators. 7 60

17 Now look at the following. It can be done several equivalent ways. 6 8 or In the first eample I used what is called the least common denominator (the smallest number which is divisible by both denominators.) while the second time I just multiplied each fractions top and bottom number by the denominator of the other fraction. This second way required the etra step of reducing the fraction so it seem like an odd way to do it. In truth it does not matter which way you do it. Sometimes the etra time it would take to figure out what is the least common denominator is more than it takes to reduce the fraction which sometimes has to be done even if you did find the least common denominator. Subtraction The same as addition ecept that you subtract the numerators after you get a common denominator. ***Please be aware that I epect you to be able to do addition and subtraction on your own so throughout this year I epect to always see the work (ie, showing the fractions with common denominator before you write the final result) Multiplication This is usually a lot easier than addition or subtraction. Addition and subtraction require the somewhat messy and frequently error prone process of common denominators. Multiplication does not require this. For multiplication you just need to multiply across numerator times numerator and denominator times denominator. So here is a typical case: While at times it is necessary to reduce the final outcome, it really never gets more difficult than that. Here is a number of eamples just to justify having a section on multiplication

18 Note that you must always reduce your final answer!!! Also, never leave anything as a number over one in your final answer you write 6/ as just 6. Please note that it would have been very easy to interpret the as times / which is why mied numbers should never show up in problems ecept as a final answer. Division Division is relatively simple once you know the trick. I actually covered it earlier but I thought this would be a good time to point out a few things. Remember that division can be rewritten as just the multiplication by the reciprocal. Look at the following eample of how this is done. 7 7 A few things are worth pointing out. First note that only the second of the pair (the one being divided by) is flipped over. The first number is left alone. It is not a case of just flip one over and it does not matter which. You always leave the first alone and take the reciprocal of the second. Of course you may already be aware that division shows up in different forms so take a look at this means 8 8 Again make sure you always reduce your final answer as much as possible. P - P P - P - 7 P - 7 P P -7 7 P -8 7 P P -0 P - 9

19 Unit : Single Transformation We will first deal with simple cases in which only one operation is required to solve the equation. Baring the use of more advanced functions, there are only four types that we need to worry about: addition, subtraction, multiplication, and division. In a sense however there is not even that many different things to worry about. As we will see, each case boils down to finding a way to cancel out the unwanted part. We then need to make sure that we do the same thing to both sides. By doing this we are assured that even though we are changing the appearance of the equation we are not actually changing situation. For eample, suppose we had the rather ridiculous equation of =. It is obviously true so hopefully no arguments so far. Now if I were to add three to just one side we would get + = or 7 = which of course makes no sense whatsoever. If we epect to get an equation that makes sense we need to add the same things to both sides. + = + to get 7 =7. This idea of doing the same things to both sides is usually referred to as keeping the equation balanced. Addition +a=b Our goal here is get the variable by itself. In most cases, at least in the beginning, the a and b are actually just numbers. For eample, + = 0 is a typical problem. One way to think about the solving of this problem is to ask what cancels a +. Well since the - = 0, the answer is a and the solving of the problem is done to match Here is another eample 7 In reality it does not matter what the number being added to. It does not have to even be a definite number. Even if it is just a variable the method of solution is still the same. a 0 a a 0 a 0 a While we can not actually perform the operation of 0 a, we have solved the equation for which was our goal. What if the equation looks a little different - such as + =9. We solve it same way although it may help to remember the basic rule, the Communitive property of addition which says a + b = b + a or + = +. So here is the solution to our problem Actually we rarely ever visibly switch the three and around but until you are confident about what you are doing, feel free to actually do so.

20 Subtraction -a=b (some of the wording in this section will probably seem familiar) Once again we want to get by itself and again the a and b are usually just numbers. The only real difference is that now we are forced to ask what cancel out a minus number. Lets use + = 0 as our typical problem. Well since and are additive inverses (i.e. - + = 0), the answer is to add. 0 Here is another eample 0 7 Again it does not matter what the number being subtracted from. It does not have to even be a definite number. Even if it is just a variable the method of solution is still the same. a 0 a a 0 a 0 a While we can not actually perform the operation of 0+ a, we have solved the equation for which was our goal. What if the equation looks a little different - such as = 9. The last time we used the Communitive property to switch the and around but we can not do it here since there is no Communitive property of subtraction. Why not? Well look at this: 9 = 6 while 9 = -6. We get the opposite number if we try to switch them around. We will deal with solving this type of problem later since it is possible to rewrite the problem as a two step problem( + -. =9) and we only wanted to handle single step problems for now. Hold on a second! You probably did not realize it yet but I glossed over a type of problem in both of the last two sections. What if you have + - = 8. There are actually two ways of handling this problem. You could handle it like an addition problem. We basically learned there to subtract the number that was being added and that works That may seem kind of messy so most people prefer to remember the definition of subtraction which says that = + - ecept we will use it in reverse

21 8 8 8 Which way you deal with it is up to you. Just make sure you whatever you have on the left side does cancel out to give just ( or m or p or whatever variable is used. Leave eact answers (use decimals only if they are eact and reduce all fractions) Do these problems without a calculator and as always show your work Solve for the variable -- if more than one solve for P = P - = P = P - r 0 = 0 P - + = P = P -7 b+ ½ = P -8 9 P -9 P = 6 P - + a = 6 P - b = d Multiplication. = In some ways this is actually easier. What is the opposite of multiplying by dividing by three is the somewhat natural answer so here is the solution. Of course it is my nature to complicate things so I am also going to offer an alternative solution method. Remember multiplicative inverses, otherwise known as reciprocals? They make some problems easier. First lets redo the last one. So why bother with this. Look at the following problem

22 This is basically a mess. Try it this way now Not a big improvement but, once you are used to it, it does get better at least in cases involving fractions. By the way note that there is a Communitive property for multiplication so solving. = is the same as solving =. Either way you divide by. Division a=b or b a Well this is actually pretty simple since what is the opposite of dividing something into pieces putting it together. Say what? Well do not worry about that but just think for a moment. What is the opposite of dividing? The answer is of course multiplying. 6 of course the same problem could have been written differently but the solution is the same 6 Please note that = is a very different problem and just like - is not a one step problem

23 Summary So to review single operations here is a quick chart type eample idea shown answer 9 6 I can get by just knowing to always do the opposite operation with the same number as you start with. You can also use the same operation with the opposite number (meaning the inverse additive for addition or subtraction and multiplicative, reciprocal, for multiplication or division Same directions as last ones P - m. = P - p = P - = P P -7 8 P -8 w P -9 t=k P -0 a= P - b =p P - Demonstrate both ways of solving this problem -- using division and multiplying by reciprocal

24 Unit : Solving Equation complications Last time we just dealt with very simple equations. All you had to do was one simple operation in order to solve it. This time we need to look at slightly more complicated problems. For now we will still avoid more comple matters such as higher powers and square roots but we do need to deal with cases that involve two or more operations. It may be an operation on one side or it may be that two things need to be both sides rather than just one. Combing Like Terms At times it may be that you need combine terms on one side before dealing with crossing values from one side to the other. You probably, or at least hopefully, referred to terms which could be combined together as like terms. A term is just a less fancy name for what is technically called a monomial. A monomial consists of elements joined only by multiplications. It could be just a number, a variable, or maybe both. It may even be the case that there are multiple variables in a monomial. It may be repetitive but it is important to say again that it can not have any operation other than multiplication. Here are some eamples to clarify. these are monomials,,,, 7, 9y, y z w, these are not monomial - +, /, 9, + y Monomial terms are said to be like terms if they have the eact same variables and each variable has the same eponent. This means that and 8 are like terms since both have only and each has a power or. We can therefore take the epression + 7 and say it equals. On the other hand, if we were given + 7 or + 7y then there is nothing you can do with them. In the first case, the terms each have as the only variable but the eponents do not match. The second case has eponents of for both terms but one has and the other y. Even when people understand the idea of matching up variables they often make one mistake. There is a tendency to say that + equals 9 even though and do not have the same variables and thus are not like terms. In fact it is easy to prove that this did not work. To test if an operation does or does not work we can simply try a value for. In this case, lets us equals. So + = + () = + 8 = while 9 = 9() = 8. While it is possible for one particular value to appear to work, any time we find any value of which shows the two are not equivalent than we know it is not correct. Be warned that if the two had come out to the same thing we could only say we have evidence which suggests that it might be true while even one countereample proves something is false. So it is about time we get some actual problems It is hardly any different than what we had in the last unit. There is just the etra step of adding to 7.

25 Two or more transformations The real focus of this packet is on solving equations for with two or more transformations are required to solve it. To make sure you know what I am talking about, look at the following 6 8 If it were just = 8 you would divide by three. If it were just + 6 = 8 you would subtract 6. Since both are there, you need to actually do both operations in order to solve the problem. The only question is which do you do first? Imagine for a moment that you had +6 and had just learned =. You would then have. +6. According to the order of operations, to simplify this you would multiply the by the and then add 6. In some ways you can think of solving equations as going through the order of operations in reverse order. You deal with the 6 and then with the. Eample : first subtract 6 from both sides divide both by Another way of thinking about it is to always deal with the part least closely connected to the variable we are solving for first. Eample : 9 9 first subtract 6 from both sides divide both by Eample : Eample : 0 6 7( ) divide by 7 subtract divide by

26 This last eample also demonstrates the minimum amount of work you should be showing. What your rules were during Algebra is hard to say but at this point you can start showing a little less. You do not need to actually write the s when subtracting from each side. You just show a new line where the four has been subtracted. You should however note that you still do one step at a time. No subtracting by and dividing by three in the same step. Eample also shows something for you to think about. You may have noticed that you could have distributed the seven through the parenthesis and then dealt with +8=. This works about the same. It does not matter which you do. In fact, if the seven did not divide evenly into the than it would have been better to distribute through. Which Side? There will be times when you are not sure which way to do things because you will have on both sides. The answer of which side to work with is really easy it does not matter. Although we prefer that final answers place on the left (such as = ), it does not matter which way you solve it. The idea is just to make sure all s (or whatever other variable you are solving for) are on one side while all other numbers and variables are on the other. For eample, suppose you are given + = 9 6 You actually have quite a few options on problems like this. Lets suppose that you decide to move s to the left first. You subtract 9 from each side to get -+=-6 At this point it is just like the problems from the last section, a two transformation solving. Here is the full solution Eample : Of course you would end up with the same thing if you had subtracted from each side and then solved the problem = 6. Some people always prefer to solve by putting the variable on the left side while others say to always put things to the side where there are more s to start with (in this case the 9 side). It is really up to you The keys to really remember is that, unless there are parenthesis involved, you should always deal with adding and subtracting items prior to multiplying or dividing addsub before multdiv. Here are some additional eamples:

27 Eample 6: Eample 7: ( ) / 7 (7 ) ******* Note that in eample 7 I dealt with getting rid of the division first. When there are parenthesis, you must get rid of what is outside the parenthesis first wither by distributing through or canceling it to the other side. You may have also noticed that I switched the final answer around to that the is on the left. Finally notice that I did I step at a time only. You are epected to show one step for moving the s and another one for moving the numbers. Eample 8: ( ) / 7 or or. a-=b This is a case I mentioned in the last packet. It is frequently handled incorrectly so be careful. It is a good idea to remember the definition of subtraction before tackling this problem a- = b a + - = b So say you are given the problem of = You may not have to actually rewrite the problem but you should think of it as + - = With this idea in place the first step is just like normal subtract form each side + - = or

28 - = - The negative in front of can be cancelled out by multiplying by or dividing by by (they are really the same thing!) = Here is another eample without the commentary 9 9 It is a common mistake to subtract the 9 and then just forget about the - but hopefully you will avoid this. It is a really annoying way to loose points. Check your answers If you have not been checking your answers thus far, this is a really good place to start. Simply take whatever you found to be the answer and put it in for. For eample, in the last problem = - so put in for 9 = 9- - = 9 + = = Yes, we were right! Checking your answer is a simple quick process which will catch a lot of errors. You do not even have to write it down. It is usually all numbers so you can often just do it in you head. Now I realize that on a test you may be pressed for time but for simple problems like this you will usually be better served by catching you answers than making sure you get to all the problems if that means leaving errors. Of course if you want to just get through the test, go back when you finish and check and recheck your answers. It is actually a valid argument to point out that if you do your check right away you may make the same mistake again (such as saying + = 6) If you come back a few minutes later you may catch this mistake, Problems solve each problem showing all your work If there is more than one variable, solve for P = 6 P = P = 8 P - + = P - a + = P (+) = 9 P -7 9

29 P -8 ( ) = 9 P -9 = 0 P = P = + 7 P = a P a = P - 8 = P = 8 P = P -7 6(+(+)) = 6 P (hint if you think of it as [] +=0 what do you do first? P -9 9 (hint find the least common denominator and multiply both sides by it) P = might as well have an easy one again

30 Unit 6: Decimals and graphs Decimals and Fractions It happens that occasionally in solving a problem that you end up with something like this..7 This is absolutely not an acceptable answer. A basic rule to always remember is that you never leave both fractions and decimals in your answer. Of course many of you reach for your calculators but you should have learned my feeling about those by now but lets say you did anyway. You would get.7879 There is no eact decimal answer for this and unless I specifically said to, you are not supposed to round your answers off. So how do you get an eact answer? The method of handling this problem is actually pretty simple. Look at the problem. How many decimal places does it have at most? So that means to multiply top and bottom by 00. After that we can reduce if possible Keep in mind that we need to multiply both top and bottom by the same thing. Now if you look at this problem you may realize that it would have been possible to multiply by something less than one hundred since we then had to reduce. In this particular case we could have multiplied by 0 but the time it takes to figure out what the best choice is probably more than it took to just use one hundred and then reduce. Lets quickly go through another eample Solving Equations with decimals There is actually a way to avoid having decimals show up in fractions while solving. The answer is to avoid decimals in the first place. Of course you can not just skip these problems since, with our monetary system utilizing two decimal places, there are too many important problems you could not do. What you can do is handle the decimals first. Take this problem, Similar to what we did last time, look for the most decimal places which in this case is two. This again means to multiply by 00 but rather than both top and bottom we do it to both sides. 00(..) Now we can solve it like we did in the last packet which I leave up to you to do. The key to this is to find the most decimals places of any part and then multiply by that. Be careful to not try to apply this rule to cases where it is impossible to do so. For eample, it does not work for just an epression like If I were to multiply by 00 it would be changing the value of the epression. The only time I can ever introduce an operation is when

31 there are two sides (in other words an equation). Trying to do it on an epression is like saying that = 00. = 00 which is of course crazy. You may also run into situations like this.. (..9).9 It is probably best to deal with this in one of two ways..(..9).9 or by (..6.9) 6 90.(..9).9 0.(..9) 0.9 (..9) (..6) My preference is for the first but either way just make sure you realize that you can not multiply both the number outside the parenthesis and the numbers inside. A simple eperiment should convince you that it does not work. Graphing One Variable Equations Well it is time we dealt with some graphing problems but we will start with some simple problems. Equations with only one variable But before we do lets go over some basic ideas for graphing in dimensions. ) Always clearly label the scale on you graphs ) Label any important points especially those used in graphing ) The first number in an ordered pair is the and the second is the y ) The horizontal ais is the -ais and the vertical is the y-ais ) When in doubt plot a bunch of points Given any problem it is always possible to get a rough idea of the graph just by plotting points Take for eample y = + 9 while others may know already that this is a U shape (technically called a parabola), you may have no idea. Set up what I call a T-Table y

32 At this point just fill in some arbitrary points for (try to choose both positive and negative and always use 0. y Then put each value into the equation to see what you get. y Plot the points and see what you get. You should hopefully recognize the shape as a parabola. Of course with equations you recognize as lines it would only be necessary to plot two points but I am going to say that you must do three since that will provide a simple error check. Now back to our equations with only variable. Try graphing = and you may think I am crazy. After all that does not appear to be the equation of any graph. It is an answer not an equation to graph right? nope! A graph is merely the representation of all points which satisfy an equation. Look at the points (,-), (,0), (,0) and (,). In each case isn t it true that =. These points satisfy the equation as do all these points The graph of a = any number turns out to be a vertical line. This can be hard to remember since the ais is the horizontal one but if you just remember it is the opposite you will do okay.

33 Given that a vertical line is of the form = a, what do you suppose y = would look like? Well it of course is a horizontal line. In fact y = looks like Graphing Linear Equations with Two Variables by T- Table I have actually already talked about this actually. Since this method does not use any of the special properties of lines, all you are supposed to do is construct a T-Table using at least points and plot the points. The only things I have yet to mention is that you should make sure you put little arrows on the end of the line you drew. Please note that I said line. It is supposed to be straight ie no curves allowed. It would not be a bad idea to use a straight edge although I am not gong to require that if you can draw them to at least look straight. P =.9 P 6-. =. P = 7. P 6-.( +.9) =. P = P =. Graph the following (preferably on graph paper) ( use T tables for variable types and label your scale always) P 6-7 = 9 P 6-8 = - P 6-9 y =. P 6-0 y = 6 P 6- = P 6- y = + P 6- y = 0 P 6- y = - + P 6- y = P 6-6 y